library(bayesrules)
library(rstanarm)
## Loading required package: Rcpp
## This is rstanarm version 2.21.1
## - See https://mc-stan.org/rstanarm/articles/priors for changes to default priors!
## - Default priors may change, so it's safest to specify priors, even if equivalent to the defaults.
## - For execution on a local, multicore CPU with excess RAM we recommend calling
## options(mc.cores = parallel::detectCores())
library(bayesplot)
## This is bayesplot version 1.8.1
## - Online documentation and vignettes at mc-stan.org/bayesplot
## - bayesplot theme set to bayesplot::theme_default()
## * Does _not_ affect other ggplot2 plots
## * See ?bayesplot_theme_set for details on theme setting
library(tidyverse)
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library(broom.mixed)
library(tidybayes)
If we find a relationship between two independent variables and the dependent variable (and those two independent variables are independent of eachother), then we might want to build a model that incorporates both of their influences’ on the outcome variable.
Ford is the reference category
The difference in a Ford car’s miles per gallon and a Subaru car’s miles per gallon.
The typical miles per gallon of a Ford car.
B0 represents the typical size of a Mr Stripey tomato at 0 days of growth; B1 represents the amount a tomato increases in weight for 1 more day of growth; B2 represents the typical difference in weights between a Mr Stripey tomato and a Roma tomato at any day of age.
If B2 were 0 there would be no difference in weight between Roma and Mr Stripey tomatoes.
The relationship between tomato size and age varies depending on the type of tomato.
B3 represents the differences in the relationship between age and weight for the 2 different types of tomatoes (the different slopes that each of these relationships possess).
11.5
By adding more predictors, we can improve the predictive accuracy of our posterior model.
By removing predictors, we can better isolate the relationship between two variables.
Height. Height has been found to correlate with foot size.
Whether the child knows how to swim. I think the relationship between swimming ability and shoe size would be spurious, and would uncessarily complicate the model.
A good model produces a posterior distribution that closely matches the observed distribution; has a low MAE scaled; has a high percentage of values that fall within the within_50 interval.
A bad model has a posterior distribution that deviates strongly from the shape of the observed distribution (perhaps the wrong type of model was chosen); has a high MAE scaled; has a low percentage of values that fall within the within_50 and within_95 interval.
We want to include enough predictors that our model is accurate (i.e. closely fit to our data), but not so many predictors that our model is “overfit” (i.e. biased). Therefore, when considering how many predictor variables to include, we entertain the bias-variance tradeoff, considering the fact that we don’t want our model to be biased (overfit) nor do we want it to have too much variance (want it to be an accurate predictor).
data("penguins_bayes")
# Alternative penguin data
penguin_data <- penguins_bayes |>
filter(species %in% c("Adelie", "Gentoo")) |> drop_na()
ggplot(penguin_data, aes(x = flipper_length_mm, y = body_mass_g, color=species)) +
geom_point() +
geom_smooth(method = "lm", se = FALSE)
## `geom_smooth()` using formula 'y ~ x'
penguin_model_1 <- stan_glm(
body_mass_g ~ flipper_length_mm + species,
data = penguin_data, family = gaussian,
prior_intercept = normal(3100, 50),
prior = normal(33, 2.5, autoscale = TRUE),
prior_aux = exponential(0.001, autoscale = TRUE),
chains = 4, iter = 4000*2, seed = 84735, verbose=FALSE)
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pp_check(penguin_model_1)
Check out some draws from our model
penguin_data %>%
add_fitted_draws(penguin_model_1, n = 50) %>%
ggplot(aes(x = flipper_length_mm, y = body_mass_g, color = species)) +
geom_line(aes(y = .value, group = paste(species, .draw)), alpha = 0.1)
## Warning: `fitted_draws` and `add_fitted_draws` are deprecated as their names were confusing.
## Use [add_]epred_draws() to get the expectation of the posterior predictive.
## Use [add_]linpred_draws() to get the distribution of the linear predictor.
## For example, you used [add_]fitted_draws(..., scale = "response"), which
## means you most likely want [add_]epred_draws(...).
set.seed(84735)
predictions_1 <- posterior_predict(penguin_model_1, newdata = penguin_data)
ppc_intervals_grouped(penguin_data$body_mass_g, yrep = predictions_1,
x = penguin_data$flipper_length_mm, group = penguin_data$species,
prob = 0.5, prob_outer = 0.95,
facet_args = list(scales = "fixed")) +
labs(x = "flipper_length_mm", y = "body_mass_g")
From both of these visual diagnostics, we can see that our model follows the shape of the distribution pretty well, and that the majority of our predicted values fall within at least the 95% posterior prediction interval.
Some numerical summaries
prediction_summary_cv(model = penguin_model_1, data = penguin_data, k = 10)
## $folds
## fold mae mae_scaled within_50 within_95
## 1 1 427.1594 0.6729862 0.4814815 1.0000000
## 2 2 537.3611 0.8637755 0.4230769 0.9230769
## 3 3 536.7048 0.8241943 0.4074074 0.8888889
## 4 4 619.8548 0.9002595 0.1923077 0.9615385
## 5 5 356.5858 0.5338050 0.5555556 0.9259259
## 6 6 405.2069 0.6354953 0.5000000 1.0000000
## 7 7 405.6854 0.6270410 0.5384615 0.9615385
## 8 8 504.9042 0.7843185 0.4074074 1.0000000
## 9 9 357.7319 0.6067848 0.5769231 0.9230769
## 10 10 579.3295 0.8830854 0.3333333 1.0000000
##
## $cv
## mae mae_scaled within_50 within_95
## 1 473.0524 0.7331745 0.4415954 0.9584046
The numerical statistics confirm what the visualizations demonstrated–most (95%) of our data are within the 95% prediction interval, and close to half are within the 50% prediction interval.
tidy(penguin_model_1, effects = c("fixed", "aux"),
conf.int = TRUE, conf.level = 0.80)
The flipper_length_mm coefficient refers to the amount of increase in body_mass caused by a one unit change in flipper_length for Adelie penguins. The speciesGentoo coefficient refers to the difference in body_mass_g for a given flipper_length between the Adelie and Gentoo species.
# Simulate a set of predictions
set.seed(84735)
prediction <- posterior_predict(
penguin_model_1,
newdata = data.frame(flipper_length_mm= 197,
species = "Adelie"))
# Plot the posterior predictive models
mcmc_areas(prediction) +
xlab("body_mass_g")
The body_mass of an Adelie penguin with flipper length 197 is around 3700, with likely values ranging from 3250 to 4000.
penguin_model_2 <- stan_glm(
body_mass_g ~ flipper_length_mm + species + flipper_length_mm:species,
data = penguin_data, family = gaussian,
prior_intercept = normal(3100, 50),
prior = normal(33, 2.5, autoscale = TRUE),
prior_aux = exponential(0.001, autoscale = TRUE),
chains = 4, iter = 4000*2, seed = 84735, verbose=FALSE)
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penguin_data %>%
add_fitted_draws(penguin_model_2, n = 50) %>%
ggplot(aes(x = flipper_length_mm, y = body_mass_g, color = species)) +
geom_line(aes(y = .value, group = paste(species, .draw)), alpha = 0.1)
## Warning: `fitted_draws` and `add_fitted_draws` are deprecated as their names were confusing.
## Use [add_]epred_draws() to get the expectation of the posterior predictive.
## Use [add_]linpred_draws() to get the distribution of the linear predictor.
## For example, you used [add_]fitted_draws(..., scale = "response"), which
## means you most likely want [add_]epred_draws(...).
The slope of the line is slightly less steep than the model without the interaction term for both Adelie and Gentoo penguins. This implies that the interaction “dampens” the strength of the relationship between flipper_length and body_mass for both species. We can estimate the overall slope (including the relationship between flipper_length_mm and body_mass_g and the relationship between the interaction between flipper_length_mm/species and body_mass_g) as around 33 for Adelie penguins, and 50 for Gentoo penguins.
tidy(penguin_model_2, effects = c("fixed", "aux"),
conf.int = TRUE, conf.level = 0.80)
The coefficient for the speciesGentoo term changed significantly with the interaction term included in the regression model (from 234 to -3899) despite the fact that we used the same priors (so the only difference between the two models was the interaction term). Therefore, the interaction must have a strong effect, for when we allow the interaction to be present, the species coefficient changes in value greatly. (CHECK THIS LOGIC )
penguin_model_3 <- stan_glm(
body_mass_g ~ flipper_length_mm + bill_length_mm + bill_depth_mm,
data = penguin_data, family = gaussian,
prior_intercept = normal(3100, 50),
prior = normal(33, 2.5, autoscale = TRUE),
prior_aux = exponential(0.001, autoscale = TRUE),
chains = 4, iter = 4000*2, seed = 84735, verbose=FALSE)
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posterior_interval(penguin_model_3, prob=0.8)
## 10% 90%
## (Intercept) -7378.32923 -5480.31256
## flipper_length_mm 27.45303 37.79006
## bill_length_mm 56.63118 83.84360
## bill_depth_mm 29.00648 74.40261
## sigma 406.01564 490.09006
penguin_model_4 <- stan_glm(
body_mass_g ~ flipper_length_mm,
data = penguin_data, family = gaussian,
prior_intercept = normal(3100, 50),
prior = normal(30, 2.5, autoscale = TRUE),
prior_aux = exponential(0.001, autoscale = TRUE),
chains = 4, iter = 4000*2, seed = 84735, verbose=FALSE)
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penguin_model_5 <- stan_glm(
body_mass_g ~ species,
data = penguin_data, family = gaussian,
prior_intercept = normal(3100, 50),
prior = normal(30, 2.5, autoscale = TRUE),
prior_aux = exponential(0.001, autoscale = TRUE),
chains = 4, iter = 4000*2, seed = 84735, verbose=FALSE)
##
## SAMPLING FOR MODEL 'continuous' NOW (CHAIN 1).
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penguin_model_6 <- stan_glm(
body_mass_g ~ flipper_length_mm +species,
data = penguin_data, family = gaussian,
prior_intercept = normal(3100, 50),
prior = normal(30, 2.5, autoscale = TRUE),
prior_aux = exponential(0.001, autoscale = TRUE),
chains = 4, iter = 4000*2, seed = 84735, verbose=FALSE)
##
## SAMPLING FOR MODEL 'continuous' NOW (CHAIN 1).
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penguin_model_7 <- stan_glm(
body_mass_g ~ flipper_length_mm +bill_length_mm + bill_depth_mm,
data = penguin_data, family = gaussian,
prior_intercept = normal(3100, 50),
prior = normal(30, 2.5, autoscale = TRUE),
prior_aux = exponential(0.001, autoscale = TRUE),
chains = 4, iter = 4000*2, seed = 84735, verbose=FALSE)
##
## SAMPLING FOR MODEL 'continuous' NOW (CHAIN 1).
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pp_check(penguin_model_4)
pp_check(penguin_model_5)
pp_check(penguin_model_6)
pp_check(penguin_model_7)
prediction_summary_cv(model = penguin_model_4, data = penguin_data, k = 10)
## $folds
## fold mae mae_scaled within_50 within_95
## 1 1 357.2165 0.5639881 0.5185185 0.8888889
## 2 2 460.9673 0.6774634 0.5000000 0.9615385
## 3 3 398.3068 0.6603842 0.5185185 0.9259259
## 4 4 587.2437 0.9852940 0.3076923 0.8846154
## 5 5 510.9056 0.8015057 0.4444444 1.0000000
## 6 6 424.3274 0.6659373 0.5000000 0.9615385
## 7 7 460.8302 0.6769068 0.5000000 1.0000000
## 8 8 646.7795 1.0032701 0.2592593 0.9259259
## 9 9 432.2585 0.6410494 0.5384615 1.0000000
## 10 10 464.0117 0.6713420 0.4814815 1.0000000
##
## $cv
## mae mae_scaled within_50 within_95
## 1 474.2847 0.7347141 0.4568376 0.9548433
prediction_summary_cv(model = penguin_model_5, data = penguin_data, k = 10)
## $folds
## fold mae mae_scaled within_50 within_95
## 1 1 571.9361 0.7122460 0.4814815 1.0000000
## 2 2 923.5512 1.2350226 0.2307692 0.9615385
## 3 3 686.8489 0.8683759 0.3703704 0.9629630
## 4 4 690.1843 0.9220871 0.3846154 0.8846154
## 5 5 457.4796 0.5597983 0.5925926 1.0000000
## 6 6 793.3704 0.9469641 0.3846154 0.9615385
## 7 7 530.6584 0.6196611 0.5384615 1.0000000
## 8 8 515.8149 0.6445739 0.5185185 0.9629630
## 9 9 569.8630 0.6754785 0.5000000 1.0000000
## 10 10 540.4538 0.6566891 0.5185185 1.0000000
##
## $cv
## mae mae_scaled within_50 within_95
## 1 628.0161 0.7840896 0.4519943 0.9733618
prediction_summary_cv(model = penguin_model_6, data = penguin_data, k = 10)
## $folds
## fold mae mae_scaled within_50 within_95
## 1 1 528.3690 0.8317152 0.3703704 1.0000000
## 2 2 401.8910 0.6329174 0.5000000 0.9615385
## 3 3 435.7046 0.6584721 0.5185185 0.9629630
## 4 4 535.1367 0.8440899 0.4230769 0.9615385
## 5 5 440.9778 0.6839468 0.4814815 0.9629630
## 6 6 646.0260 0.9511811 0.4230769 0.9615385
## 7 7 465.5016 0.7201086 0.4230769 0.9230769
## 8 8 418.6512 0.6450584 0.5185185 1.0000000
## 9 9 475.2119 0.7515951 0.3846154 0.9230769
## 10 10 472.2740 0.7695869 0.4074074 0.9259259
##
## $cv
## mae mae_scaled within_50 within_95
## 1 481.9744 0.7488672 0.4450142 0.9582621
prediction_summary_cv(model = penguin_model_7, data = penguin_data, k = 10)
## $folds
## fold mae mae_scaled within_50 within_95
## 1 1 399.7241 0.8801726 0.2962963 0.8888889
## 2 2 278.6517 0.5098146 0.6923077 1.0000000
## 3 3 323.5021 0.6108105 0.5185185 0.9259259
## 4 4 373.6903 0.7309572 0.4615385 0.9615385
## 5 5 501.4144 0.9351296 0.3333333 1.0000000
## 6 6 227.3234 0.4555048 0.5769231 0.9615385
## 7 7 365.6826 0.7485364 0.4615385 0.9230769
## 8 8 426.2163 0.8059264 0.3703704 1.0000000
## 9 9 389.0706 0.7205595 0.4615385 0.9615385
## 10 10 251.0481 0.4757839 0.5925926 0.9259259
##
## $cv
## mae mae_scaled within_50 within_95
## 1 353.6324 0.6873196 0.4764957 0.9548433
# Calculate ELPD for the 4 models
set.seed(84735)
loo_1 <- loo(penguin_model_4)
loo_2 <- loo(penguin_model_5)
loo_3 <- loo(penguin_model_6)
loo_4 <- loo(penguin_model_7)
# Results
c(loo_1$estimates[1], loo_2$estimates[1],
loo_3$estimates[1], loo_4$estimates[1])
## [1] -2053.593 -2124.003 -2052.450 -1994.006
loo_compare(loo_1, loo_2, loo_3, loo_4)
## elpd_diff se_diff
## penguin_model_7 0.0 0.0
## penguin_model_6 -58.4 5.8
## penguin_model_4 -59.6 5.6
## penguin_model_5 -130.0 9.6
First, let’s look at the relationship between bill_length and bill_depth
penguin_data <- penguin_data |> drop_na()
ggplot(penguin_data, aes(x = bill_depth_mm, y = bill_length_mm)) +
geom_point() +
geom_smooth(method = "lm", se = FALSE)
## `geom_smooth()` using formula 'y ~ x'
Now, let’s check out the relationship with body_mass
penguin_data <- penguin_data |> drop_na()
ggplot(penguin_data, aes(x = body_mass_g, y = bill_length_mm)) +
geom_point() +
geom_smooth(method = "lm", se = FALSE)
## `geom_smooth()` using formula 'y ~ x'
Building three penguin models
penguin_model_8 <- stan_glm(
bill_length_mm ~ bill_depth_mm,
data = penguin_data, family = gaussian,
prior_intercept = normal(50, 5),
prior = normal(-1.5, 0.005, autoscale = TRUE),
prior_aux = exponential(0.001, autoscale = TRUE),
chains = 4, iter = 4000*2, seed = 84735, verbose=FALSE)
##
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penguin_model_9 <- stan_glm(
bill_length_mm ~ body_mass_g,
data = penguin_data, family = gaussian,
prior_intercept = normal(35, 5),
prior = normal(0.005, 0.005, autoscale = TRUE),
prior_aux = exponential(0.001, autoscale = TRUE),
chains = 4, iter = 4000*2, seed = 84735, verbose=FALSE)
##
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penguin_model_10 <- stan_glm(
bill_length_mm ~ bill_depth_mm+body_mass_g,
data = penguin_data, family = gaussian,
prior_intercept = normal(50, 5),
prior = normal(0, 0.005, autoscale = TRUE),
prior_aux = exponential(0.001, autoscale = TRUE),
chains = 4, iter = 4000*2, seed = 84735, verbose=FALSE)
##
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pp_check(penguin_model_8)
pp_check(penguin_model_9)
pp_check(penguin_model_10)
prediction_summary_cv(model = penguin_model_8, data = penguin_data, k = 10)
## $folds
## fold mae mae_scaled within_50 within_95
## 1 1 2.261209 0.5042860 0.7037037 0.9259259
## 2 2 3.315736 0.7566139 0.4615385 0.9230769
## 3 3 2.326412 0.5135774 0.6296296 0.9629630
## 4 4 2.130073 0.4711664 0.6923077 1.0000000
## 5 5 2.649268 0.5910488 0.5925926 0.9629630
## 6 6 4.275919 0.9758033 0.3076923 0.9615385
## 7 7 3.082429 0.6899118 0.5000000 0.9615385
## 8 8 3.273684 0.7224448 0.4444444 1.0000000
## 9 9 2.805641 0.6343910 0.5384615 0.9615385
## 10 10 4.601239 1.0966927 0.3703704 0.8888889
##
## $cv
## mae mae_scaled within_50 within_95
## 1 3.072161 0.6955936 0.5240741 0.9548433
prediction_summary_cv(model = penguin_model_9, data = penguin_data, k = 10)
## $folds
## fold mae mae_scaled within_50 within_95
## 1 1 1.685671 0.6685488 0.5185185 0.9629630
## 2 2 1.234169 0.4742710 0.6923077 1.0000000
## 3 3 1.886409 0.7328654 0.4444444 0.9629630
## 4 4 1.444000 0.5696574 0.6153846 0.9615385
## 5 5 1.729281 0.6861726 0.4814815 0.8888889
## 6 6 1.554756 0.6106289 0.5384615 0.9615385
## 7 7 1.210328 0.4717706 0.6153846 0.9615385
## 8 8 2.184668 0.8537932 0.4444444 1.0000000
## 9 9 2.037541 0.7879526 0.3461538 1.0000000
## 10 10 1.723806 0.6865360 0.4444444 0.9629630
##
## $cv
## mae mae_scaled within_50 within_95
## 1 1.669063 0.6542197 0.5141026 0.9662393
prediction_summary_cv(model = penguin_model_10, data = penguin_data, k = 10)
## $folds
## fold mae mae_scaled within_50 within_95
## 1 1 4.167887 0.7877940 0.4444444 1.0000000
## 2 2 3.566254 0.6726162 0.5000000 1.0000000
## 3 3 5.620458 1.0802408 0.3333333 0.9259259
## 4 4 4.081782 0.7712352 0.4615385 1.0000000
## 5 5 4.318533 0.8194117 0.3703704 1.0000000
## 6 6 3.639383 0.6973355 0.5000000 0.9615385
## 7 7 4.918917 0.9788260 0.2307692 0.9615385
## 8 8 3.594653 0.6806221 0.4814815 1.0000000
## 9 9 4.047367 0.7763710 0.3846154 1.0000000
## 10 10 3.939797 0.7596770 0.4444444 0.9259259
##
## $cv
## mae mae_scaled within_50 within_95
## 1 4.189503 0.802413 0.4150997 0.9774929
As we can see from both the pp_check plots and the error statistics, the third model is significantly worse than the first two, likely due to the fact that the priors for the first two models were inferred from the data, while for the last model the prior was not able to be inferred from the data. Of the first two models, I prefer the second, for the posterior predictive distribution better matches the observed distribution, and the error statistics are smaller.
Downloading weather data
data("weather_perth")
# Alternative penguin data
weather_data <- weather_perth |> drop_na()
First, check out the data
ggplot(weather_data, aes(x = windspeed9am, y = windspeed3pm)) +
geom_point() +
geom_smooth(method = "lm", se = FALSE)
## `geom_smooth()` using formula 'y ~ x'
Building weather model
weather_model <- stan_glm(
windspeed3pm ~ windspeed9am,
data = weather_data, family = gaussian,
prior_intercept = normal(10, 2),
prior = normal(0.4, 0.005, autoscale = TRUE),
prior_aux = exponential(0.001, autoscale = TRUE),
chains = 4, iter = 4000*2, seed = 84735, verbose=FALSE)
##
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pp_check(weather_model)
prediction_summary_cv(model = weather_model, data = weather_data, k = 10)
## $folds
## fold mae mae_scaled within_50 within_95
## 1 1 3.238853 0.6902631 0.49 0.93
## 2 2 3.951084 0.8368496 0.42 0.98
## 3 3 3.277449 0.6917662 0.49 0.95
## 4 4 3.402771 0.7201394 0.48 0.95
## 5 5 3.484423 0.7303913 0.43 0.97
## 6 6 4.065505 0.8732405 0.36 0.95
## 7 7 3.849420 0.8268458 0.43 0.93
## 8 8 3.524971 0.7341515 0.46 0.98
## 9 9 3.373769 0.7124051 0.48 0.95
## 10 10 2.701749 0.5648839 0.53 0.97
##
## $cv
## mae mae_scaled within_50 within_95
## 1 3.486999 0.7380936 0.457 0.956
The pp_check illustrates that the posterior predictive disribution pretty closely maps onto the observed distribution. The error statistics confirm this: the MAE scaled is 0.736 and around 46% of data values are within the 50% posterior predictive interval. Therefore, I’d say the model is pretty good, with the caveat that the prior values were inferred from the data, so it is at risk of overfitting.